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A note on the matrix arithmetic-geometric mean inequality
No full textThis note proves the following inequality: If for some positive integer , then for any positive definite matrices \bA_1,\bA_2,\dots,\bA_n, the following inequality holds: \begin{equation*}\label{eq:main} \frac{1}{n^3} \, \Big\|\sum_{j_1,j_2,j_3=1}^{n}\bA_{j_1}\bA_{j_2}\bA_{j_3}\Big\| \,\geq\, \frac{(n-3)!}{n!} \, \Big\|\sum_{\substack{j_1,j_2,j_3=1,\\\text{, , all distinct}}}^{n}\bA_{j_1}\bA_{j_2}\bA_{j_3}\Big\|, \end{equation*} where represents the operator norm. This inequality is a special case of a recent conjecture proposed by Recht and R\\u27{e} (2012)
On n/p-Asymptotic Distribution Of Vector Of Weighted Traces Of Powers Of Wishart Matrices
No full textThe joint distribution of standardized traces of and of , where the matrix follows a matrix normal distribution is proved asymptotically to be multivariate normal under condition . Proof relies on calculations of asymptotic moments and cumulants obtained using a recursive formula derived in Pielaszkiewicz et al. (2015). The covariance matrix of the underlying vector is explicitely given as a function of and
Some Graphs Determined By Their Distance Spectrum
No full textLet be a connected graph with order . Let be the distance spectrum of . In this paper, it is shown that the complements of and are determined by their -spectrum. Moreover, it is shown that the cycle ( odd) is also determined by its -spectrum
Inertia sets allowed by matrix patterns
No full textMotivated by the possible onset of instability in dynamical systems associated with a zero eigenvalue, sets of inertias \sn_n and \SN{n} for sign and zero-nonzero patterns, respectively, are introduced. For an sign pattern \mc{A} that allows inertia , a sufficient condition is given for \mc{A} and every superpattern of \mc{A} to allow \sn_n, and a family of such irreducible sign patterns for all is specified. All zero-nonzero patterns (up to equivalence) that allow \SN{3} and \SN{4} are determined, and are described by their associated digraphs
Ordering cacti with signless Laplacian spread
No full textA cactus is a connected graph in which any two cycles have at most one vertex in common. The signless Laplacian spread of a graph is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the associated signless Laplacian matrix. In this paper, all cacti of order n with signless Laplacian spread greater than or equal to n − 1/2 are determined
